In particular, let \(\mathfrak {g}\)  be a complex Lie algebra. Let \(A=\mathcal {U}(\mathfrak {g})\)  be the enveloping algebra of \(\mathfrak {g}\) , with the standard filtration given by \(A_{n}=\mathcal {U}_{n}(\mathfrak {g})\) , the subspace of \(\mathcal {U}(\mathfrak {g})\)  spanned by products of at most \(n\) -elements of \(\mathfrak {g}\) . By the Poincaré-Birkhoff-Witt theorem (see Knapp [1]}), the graded algebra \(\text{gr} (\mathcal {U}(\mathfrak {g}))\)  is canonically isomorphic to the symmetric algebra \(S(\mathfrak {g})\) . Suppose \(M\)  is a \(\mathcal {U}(\mathfrak {g})\) -module generated by a finite-dimensional subspace \(M_{0}\) . We set \(M_{n}=\mathcal {U}_{n}(\mathfrak {g})M_{0}\) . Denote \(\text{gr}M=\bigoplus \limits _{n=0}^{\infty }\text{gr}_{n}M\) , where \(\text{gr}_{n}M=M_{n}/M_{n-1}\) . Then \(\text{gr}M\)  becomes a graded \(S(\mathfrak {g})\) -module. We denote \(\dim (M_{n})\)  by \(\varphi _{M}(n)\) . Then we have the following lemma.
